Calculating the Probability of a Shared Birthday among n People

[buddingscientist]

Suppose a room contains n people. Assuming that days of the year are equally likely to be birthdays for each person, calculate the probability that at least two of the people have a common birthday.

well I have the answer but I'm just curious as to the thought processes you go through to answer it.


what I've done (incorrectly) is:

if n = 2, P = 1/365^2
if n = 3, P = 1/365^2 (person 1, 2) + 1/365^2 (person 1, 3) + 1/365^2 (person 2, 3)

and generalized it to:
( 3 + (n sum i=3) (i)! / (i-1)! ) / 365^2

there's no question that it's wrong but that's the process I take

 

[matt grime]

work out the probability they all have different birthdays and subtract that from 1. Obviously if there are 366 people in the room the probabilit must be 1, mustn't it? (excludeing leap year things). Does n=366 give P=1 in your answer?

 

[buddingscientist]

probably not (npi), I know my answer is wrong, you recommend using conjugates (1 - P(all different)), I'll work on it some more, thanks for your response

 

[OneSquared]

Ignoring leap years:

P(bd2 = bd1) = 1/365
P(bd2 != bd1) = 364/365
P(bd3 = bd1) + P(bd3 = bd2) = 2/365, given that bd1 != bd2
P(bd3 != bd1) AND P(bd3 != bd2) = 363/365

Therefore P(bd1 != bd2 != bd3) = 1 - (363/365) * (364/365)

So for any n,
P(two birthdays are the same) =
1 - (((365-(n-1))*(365-(n-2))...(364))/(365^n))